Wavefunction orthonormal

Assume that all n electron wavefunctions orthonormal—that is, that the overlap integrals involved in the secular equation Eq. (3.8.4) are replaced by Kronecker deltas. 

Multiplying a molecular orbital function by a or P will include electron spin as part of the overall electronic wavefunction i /. The product of the molecular orbital and a spin function is defined as a spin orbital, a function of both the electron s location and its spin. Note that these spin orbitals are also orthonormal when the component molecular orbitals are. 

There are a few interesting points about the treatment. First of all, there is no variational HF-LCAO calculation (because every available x is doubly occupied) and so the energy evaluation is straightforward. For a wavefunction comprising m doubly occupied orthonormal x s the normalizing factor N is... 

More fundamentally, what Pecora seems to assume - although never explicitly saying so - is the following property. Since the condition CC+ = In is actually the orthonormalization constraint on the scalar product between any two wavefunctions (ft is hermitian. That is to say, it is assumed that the subspace on which the projection is made is a Hilbert subspace. 

A further advantage of using Lagrangian dynamics is that we can easily impose boundary conditions and constraints by applying the method of Lagrangian multipliers. This is particularly important for the dynamics of the electronic degrees of freedom, as we will have to impose that the one-electron wavefunctions remain orthonormal during their time evolution. The Lex of our extended system can then be written as ... 

Kohn and Sham provided a further contribution to make the DFT approach useful for practical calculations, by introducing the concept of fictitious non-interacting electrons with the same density as the true interacting electrons [8]. Non-interacting electrons are described by orthonormal single-particle wavefunctions y/i (r) and their density is given by ... 

However, although the locally scaled transformed wavefunctions preserve the orthonormality condition, they fail to comply with Hamiltonian orthogonality. Of course, one can recombine the transformed wavefunctions so as to satisfy the latter requirement, by solving once more the eigenvalue problem... 

Returning to equation (25), evaluation of the total vibrational overlap integral, (Xj X7), is less formidable than it appears. The vibrational wavefunctions are a complete orthonormal set for which ( 1 0 )= where S is the Kronecker delta. For the vast majority of normal modes, S (and AQe) = 0. For these modes the vibrational overlap integrals become (yjy,/) = 1 if v = v, and = 0 if v v . Except for the requirement that the vibrational quantum number must... 

Finally, some remarks will be made concerning the dimension of wavefunctions. The bound-state orbitals are subject to the orthonormality relation... 

In order to obtain the eigenvectors, restrictions from the requirement of orthonormality for the wavefunctions lP (r) have to be also incorporated. This will be demonstrated for the case of a two-state system defined in equ. (7.86). Here these restrictions lead to the additional conditions... 

As Dewar points out in ref. [30a], this derivation is not really satisfactory. A rigorous approach is a simplified version of the derivation of the Hartree-Fock equations (Chapter 5, Section 5.2.3). It starts with the total molecular wavefunction expressed as a determinant, writes the energy in terms of this wavefunction and the Hamiltonian and finds the condition for minimum energy subject to the molecular orbitals being orthonormal (cf. orthogonal matrices, Section 4.3.3). The procedure is explained in some detail in Chapter 5, Section 5.2.3)... 

To obtain eqs. (7.2.4) and (7.2.5) from eq. (7.2.6), we only need to set angle a to be 90° and 0°, respectively. It is not difficult to show that the five hybrids form an orthonormal set of wavefunctions. The parameter a in the coefficient matrix in eq. (7.2.6) may be determined in a number of ways, such as by the maximization of overlap between the hybrids and the ligand orbitals, or by the minimization of the energy of the system. In any event, such procedures are clearly beyond the scope of this chapter (or this book) and we will not deal with them any further. 

The above unconstrained optimization cannot be directly applied at intermediate values of A for which the Hamiltonian contains the unknown potential V. We found, however, that a simultaneous determination of and V can be achieved by performing the following constrained optimization. We assume that the trial many-body wavefunction results in the electron density rcA(r) and expand both nA(r) and the ground state density n(r) in a complete and orthonormal set of basis functions fs ... 

It is straightforward also to include a core of doubly-occupied orthonormal orbitals, which may either be taken unchanged from prior calculations or optimized, simultaneously with the cip and csJ coefficients, as linear combinations of the %p. Multiconfiguration variants of the SC wavefunction may also be generated, if required, and calculations may be performed directly for excited states. 

The exact Schrodinger equation of motion, equation (1), may be equivalently Stated in a manner which shows the neglected terms arising from the assumption of the product form for the wavefunction, equation (4). The exact eigenstate Yj(x, R) is expanded in terms of the complete orthonormal set of functions y>i(x R) obtained from the solutions of the electronic equation, equation (5), in which case the nuclear wavefunctions (R) appear as the coefficients in the expansion. This procedure yields the following infinite set of coupled equations for the x (R)6... 

When the orbitals are determined in this manner, with the only restriction being the orthonormality constraint, equation (12), they yield the best possible antisymmetrized product wavefunction (i.e., a single determinant wavefunction) for the system in question since the resulting fP(x R) yields the lowest possible value for E(R). The heart of the Hartree-Fock model is to replace the detailed and accurate description of the repulsions between every pair of electrons in the system by the average field that each electron exerts on every other. This is a consequence of the one-electron or orbital basis of the model which leads to a product wavefunction. The probability, Pa(ri,r2)dridr2, that electron 1 be in some small volume element about ri when electron 2 is simultaneously in some small volume element about ra is, in a simple product-type wavefunction, given by the product of the two singleparticle probabilities,... 

PROBLEM 3.13.1. Prove Eq. (3.13.20) by expanding the trial wavefunction Ptrial in terms of a complete orthonormal basis set of the true eigenfunctions T rini. nm where H 1 lb En T. . 

In principle, it is easy to calculate the electronic energy for a wavefunction of type (18). With the assumptions that the spin functions are orthonormal and that H is spin-free, the energy expectation value is... 

The generality and importance of the above results cannot be overemphasized. The wavefunction for the valence electrons may be optimized by variation of y alone, using the effective Hamiltonian in (47) with appropriate orthonormality constraints. In practice this means that, for a function built up from orbitals r of the valence space, it is only necessary to replace matrix elements < r h a > by... 

The variational principle asserts that any wavefunction constructed as a linear combination of orthonormal functions will have its energy greater than or equal to the lowest energy (E ) of the system. Thus,... 

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